Before oiling, just to be sure, take some acetone and wipe/scrub the grips after your sanding to be sure none of the old finish is still in the wood and that all the dust is gone. If you miss a spot sanding the previous finish or it penetrated the wood, a subsequent application of oil will be blotchy.

I prefer to cut my pure tung oil 50% with plain old cheap mineral spirits (paint thinner) and apply successive thin coats with a lint free rag.
Let a coat stand about an hour, and early on in the process, you may find the grip looks like it's sweating oil here and there. Wipe this excess off and let it stand a while longer. When it feels dry, rub lightly with 0000 steel wool, dust thoroughly to remove the fine steel dust from the wood, then wipe on another coat of the dilute oil mix.
Do this about once an hour for the first day, then once a day for the next week. After several coats, a sheen will magically begin to appear. Stop the process whenever the wood looks the way you like. Let dry well, wipe on and buff a thin coat of quality paste wax. I often finish furniture I build this way. You will love the look and feel of the wood.